Problem 16 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ 4 x-8=16 $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Add 8 to Both Sides

Start by eliminating the constant term on the left side of the equation. You can do this by adding 8 to both sides. This simplifies the equation to: \[ 4x - 8 + 8 = 16 + 8 \] Which becomes: \[ 4x = 24 \]

Step 2: Divide by 4

Now, isolate \(x\) by dividing both sides of the equation by 4. This will solve for \(x\): \[ \frac{4x}{4} = \frac{24}{4} \] Simplifying gives: \[ x = 6 \]

Step 3: Check the Solution

It's important to verify that our solution is correct by substituting \(x = 6\) back into the original equation. Start with: \[ 4x - 8 = 16 \] Substitute \(x = 6\): \[ 4(6) - 8 = 24 - 8 = 16 \] The left side equals the right side, confirming that \(x = 6\) is indeed the correct solution.

Key Concepts

Solving Equations

Solving linear equations is a fundamental skill in algebra that helps in finding the value of the unknown variable. In this exercise, the equation given was \(4x - 8 = 16\). To find the value of \(x\), we needed to isolate it. This is done by performing inverse operations, which means doing the opposite of what is currently applied to \(x\).

For example:

• To eliminate a subtraction, we add the same number to both sides of the equation. When we added 8 to both sides of the given equation, the \(-8\) on the left canceled out.
• Next, to eliminate a multiplication, we divide. Once we isolated \(4x\), we divided both sides by 4 to solve for \(x\), giving us \(x = 6\).

Understanding these steps is crucial because they can be applied to any linear equation, no matter the complexity. Practice these steps with different numbers to become more confident in solving equations.

Equation Verification

Verifying the solution means checking whether our calculated value satisfies the original equation. This step ensures no arithmetic mistakes were made. Once we found \(x = 6\), we needed to prove that this solution worked by putting it back into the original equation:

Start with the original equation: \(4x - 8 = 16\). Then substitute \(x\) with 6. This gives us \(4(6) - 8\).

• Calculate inside the parentheses first: \(4 \times 6 = 24\).
• Then perform the subtraction: \(24 - 8 = 16\).

The left side equals the right side. This verification not only confirms the solution is correct but also strengthens your confidence in the steps you've applied.

Algebraic Manipulation

Algebraic manipulation involves performing operations to transform and simplify equations to find solutions. Knowing how to effectively manipulate an equation is key in algebra.

This process includes:

• Identifying the operations being applied to the variable (in this case, subtraction and multiplication).
• Cleverly choosing opposite (inverse) operations to simplify the equation. For instance, using addition to counteract subtraction and division to counteract multiplication.
• Systematically applying these operations step-by-step to narrow down the equation to \(x = 6\).

Mastering algebraic manipulation means understanding the behavior of numbers and operations, which allows you to rearrange and solve equations with ease. With practice, you'll be able to handle more complex equations and apply these techniques in real-life problem-solving situations.